Hyponormal powers of composition operators

Abstract:

Let ${T_i},i = 1,2$, be measurable transformations which define bounded composition operators ${C_{{T_i}}}$ on ${L^2}$ of a $\sigma$-finite measure space. Denote their respective Radon-Nikodym derivatives by ${h_i},i = 1,2$. The main result of this paper is that if ${h_i} \circ {T_i} \leq {h_j},i,j = 1,2$, then for each of the positive integers $m,n,p$ the operator ${[C_{{T_1}}^mC_{{T_2}}^n]^p}$ is hyponormal. As a consequence, we see that the sufficient condition established by Harrington and Whitley for hyponormality of a composition operator is actually sufficient for all powers to be hyponormal.